PRAMANA
— journal of
physics
c Indian Academy of Sciences
Vol. 59, No. 2
August 2002
pp. 243–254
Developments in quantum information processing by
nuclear magnetic resonance: Use of quadrupolar and
dipolar couplings
ANIL KUMARa;b; K V RAMANATHANb , T S MAHESHa , NEERAJ SINHAa and
K V R M MURALIa;c
a Department
of Physics; b Sophisticated Instruments Facility, Indian Institute of Science,
Bangalore 560 012, India
c Present address: Media Lab, M.I.T., Cambridge, USA
Author to whom correspondence should be addressed. Email: anilnmr@physics.iisc.ernet.in
Abstract. Use of dipolar and quadrupolar couplings for quantum information processing (QIP) by
nuclear magnetic resonance (NMR) is described. In these cases, instead of the individual spins being
qubits, the 2n energy levels of the spin-system can be treated as an n-qubit system. It is demonstrated
that QIP in such systems can be carried out using transition-selective pulses, in CH3 CN, 13 CH3 CN,
7 Li (I = 3=2) and 133 Cs (I = 7=2), oriented in liquid crystals yielding 2 and 3 qubit systems. Creation of pseudopure states, implementation of logic gates and arithmetic operations (half-adder and
subtractor) have been carried out in these systems using transition-selective pulses.
Keywords. Quantum information processing; qubit; nuclear magnetic resonance quantum computing.
PACS No. 03.67.-a
1. Introduction
Ever since the suggestion by Feynman in 1982 that quantum mechanical devices should
simulate quantum systems much more efficiently than classical computers [1], the search
for quantum algorithms has gained great impetus. Several algorithms have been developed
[2–4] which validate the premise of Feynman. Shor’s prime factorization algorithm [4]
is creating panic among cryptographers [5]. Grover’s search algorithm has given new dimension for data base search [3]. Quantum Fourier transform [6], quantum error correction
[7,8], quantum teleportation [9], quantum logic gates [6–8], quantum simulation [1,10–15],
quantum cloning [16] and Deutsch–Jozsa algorithm [2] have created a lot of excitement in
scientific circles. Efforts are being made to develop further algorithms and for experimental
realization of quantum computers. The experimental techniques being suggested include
quantum electrodynamics, trapped ions, quantum dots and nuclear magnetic resonance
(NMR) [7,8]. Of these NMR has shown the maximum progress [16–42]. Using NMR,
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people have so far demonstrated preparation of pseudopure states [20–33], implementation of logic gates and Deutsch–Jozsa algorithm using both one- and two-dimensional
NMR [17–19,28,34–37], creation of Einstein–Podolsky–Rosen (EPR) states [38], quantum state tomography [24,39,40], implementation of Grover’s search algorithm [41] and
Shor’s factorization algorithm [42].
Most of the above have been implemented by NMR of spin-1/2 nuclei coupled to each
other by spin–spin (J) couplings. The major requirement for a spin system to act as a
device for quantum information processing is qubit addressability. This is achieved in
spin-1/2 systems by the nuclei having different Larmor frequencies and finite unequal mutual couplings among all the spins. The J-coupling is mediated among nuclei through the
covalent network, and becomes vanishingly small between nuclei which are more than 4
to 5 bonds away. This imposes a natural limit on the number of qubits an NMR system
utilizing J-couplings can have. Here we describe the possibilities of utilizing dipolar couplings among spin-1/2 nuclei and quadrupolar couplings in spin-3/2 and spin-7/2 nuclei.
These couplings are averaged to zero in rapidly and isotropically reorienting molecules in
liquid state and become too large and too many in the solid state. However, for molecules
partially oriented in anisotropic media such as liquid crystals, these systems give sharp
spectral lines arising from reduced couplings. Such systems can then be treated as suitable
candidates for quantum information processing by NMR, the aim and objective being that
one can possibly increase the number of qubits in NMR quantum information processing
protocol.
2. Dipolar coupled spin-1/2 nuclei
The dipolar couplings among spin-1/2 nuclei are larger in magnitude and longer in range
compared to the spin–spin (J) couplings. Therefore such systems hold the promise of increasing the number of qubits in NMR. The feasibility of using dipolar coupled spin 1/2
nuclei for quantum information processing by NMR was first demonstrated by Yannoni et
al [43]. Later, Fung demonstrated the feasibility of using CH 3 CN molecule partially oriented in a liquid crystal [44], wherein the three dipolar coupled protons yield a 1:2:1 triplet
in the NMR spectrum. Using this system, Fung demonstrated creation of pseudopure states
and implementation of an implicit C-NOT gate. Further use of the system for computations
requires that the problem of the degeneracy of transitions belonging to different symmetric states of an equivalent H3 group is addressed. Here we describe an improved method
for creation of pseudopure states and implementation of a complete set (24) of one-to-one
reversible 2-qubit gates using transition-selective pulses. The Hamiltonian and the eigenstates in this case have a C3 symmetry, dividing the eight eigenstates into three symmetry
manifolds. The symmetric (A) manifold (figure 1a) has four eigenstates and there are a
pair of doublet (E) manifolds. The A manifold has three transitions with the intensity ratio
of 3:4:3. The transitions of the E manifold overlap with the central transition of the A manifold, yielding a 3:6:3 triplet (figure 1b). We have earlier described a method of selection
of symmetric states (SOSS) and then used the four eigenstates of the symmetric manifold
as a two-qubit system [45]. Similarly, we have used a 13 C-labeled 13 CH3 CN molecule
oriented in the liquid crystal matrix. The carbon-13 (spin-1/2) has a different Larmor frequency (1/4th of the proton) and has dipolar couplings to the protons, and this system
therefore yields eight eigenstates belonging to the symmetric (A) manifold, which can then
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Use of quadrupolar and dipolar couplings
be treated as a three qubit system [45]. We have also demonstrated SOSS for this system
[45]. After implementation of SOSS in these two systems, we have created pseudo- pure
states and implemented logic gates using one- and two-dimensional NMR methods [45].
Some of these results are summarized in the following subsections.
2.1 Preparation of pseudopure states
Certain quantum algorithms such as Shor’s factorization algorithm requires that the
quantum system be in a pure state before the implementation of the algorithm. When
only one of the eigenstates of a system is populated the system is said to be in a pure
state. This poses unrealistic experimental conditions such as zero Kelvin temperature or an
infinitely high magnetic field. It has been suggested that since all NMR results depend only
on differences in populations of various eigenstates, one can create population-distribution
such that all levels except one have equal populations. Such a state has been termed
as a ‘pseudopure state’ and gives results which are identical to the ‘pure state’ except
for the low signal/noise. Using pseudopure states, one can use NMR of molecules at
room temperatures in the usual laboratory magnetic fields for quantum information processing.
Figure 1c shows the NMR spectrum of the protons of CH3 CN partially oriented in a liquid crystal, after the selection of symmetric states (SOSS). This spectrum arises exclusively
from the symmetric states and has the intensity ratio 3:4:3 corresponding to the equal population differences between various eigenstates of the symmetric manifold, shown schematically on the right-hand side (rhs). The equal equilibrium population differences correspond
to a high-temperature (Zeeman energy kT ) and a high-field (dipolar coupling Zeeman energy) approximation, which are very well satisfied [46]. The central transition has
higher intensity compared to outer transitions due to the higher value of the matrix elements of the observation operator I x(2;3) connecting states j2i and j3i. We have labeled the
four eigenstates as 00, 01, 10 and 11 corresponding to a two-qubit system. Here we have
adopted a labeling scheme with 00 being the lowest and 11 the highest energy state. Later
in spin-7/2 case, we will show that the labeling scheme can be freely changed to suit the
logic operation at hand.
Starting from such a population distribution after SOSS, various pseudopure states can
be easily created using transition-selective π and π =2 pulses. A transition selective π pulse
interchanges the population of selected states while a π =2 pulse (followed by a gradient/homospoil DC magnetic field pulse along the Z-direction to destroy the coherences)
equalizes their populations to an average value. The j00i pseudopure state is created by
first applying a π pulse between j11i and j10i, interchanging their populations followed
by a π =2 pulse between j10i and j01i (followed by the gradient pulse), yielding the population distribution corresponding to j00i pseudopure state, as shown on the rhs of figure
1d. A π pulse on j01i and j00i then creates the population distribution given on the rhs
of figure 1e corresponding to the j01i pseudopure state. Other pseudopure states can be
similarly created as shown in figure 1. The spectra obtained using a small angle pulse at
the end of these operations reflect the new population distribution confirming the creation
of the pseudopure states.
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Figure 1. (a) Energy level diagram showing quartet (A) and doublet (E) manifolds
of oriented CH3 CN system and the two-qubit labeling scheme of the A manifold, (b)
the equilibrium 1 H spectrum, (c) the spectrum after SOSS, and spectra corresponding
to different pseudopure states (d) j00i, (e) j01i, (f) j10i, and (g) j11i. The states and
representative relative deviation populations are given on the rhs. The pseudopure states
are prepared using the spatial averaging technique. The pulse sequences for preparing
pseudopure states are as follows: (d) π (10 $ 11) π =2(01 $ 10)G, (e) state of (d)
followed by π (00 $ 01)G, (f) state of (g) followed by π (11 $ 10) G, and (g) π (01 $
10) π =2(00 $ 01)G.
2.2 Logic gates
In the two-qubit system of the symmetric states of the oriented CH 3 CN molecule, various
logic gates have been implemented after SOSS, using transition-selective pulses. It may
be mentioned here that since a spin is not a qubit, one can not use the coupling evolution
method using spin-selective pulses, as is generally the practice for weakly coupled spin1/2 nuclei [18,19]. However, the algorithm of transition-selective pulses which we have
followed in several earlier works [17,28,37] is valid and works extremely well. For example, starting from an equilibrium population distribution after SOSS (shown in figure 2a
which is also labeled as NOP, No OPeration gate), a NOT1 gate (index 1 indicates NOT
operation on first qubit with second qubit remaining unchanged) requires π (non-selective)
π (00 $ 01) π (10 $ 11) (figure 2b); NOT2 requires π (00 $ 01) π (10 $ 11) (figure 2c);
NOT on both qubits requires a non-selective π pulse (figure 2d). Figure 2 contains a complete set of reversible one-to-one 2 qubit gates. The pulse scheme for various gates are
given in ref. [45].
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Use of quadrupolar and dipolar couplings
Figure 2. 1 H spectra of oriented CH3 CN corresponding to a complete set of 2qubit one-to-one gates implemented (after applying SOSS) using transition-selective
rf pulses. The unitary operator and pulse sequence corresponding to each gate are given
in [45].
3. Use of quadrupolar (I = 3=2 and I = 7=2) nuclei
A qudrupolar nucleus of spin I has 2I + 1 energy levels and 2I single quantum transitions which are equispaced if only the first-order quadrupolar interaction is operative. For
molecules partially oriented in anisotropic media, such as liquid crystals, such a condition is often satisfied. If 2I + 1 = 2 n , such a system can be treated as an n-qubit system
[47,48]. One of the advantages of such systems and the dipolar coupled systems over
J-coupled systems is that the coupling values are one to two orders of magnitude larger, allowing the use of correspondingly shorter transition selective pulses, with greater ease and
shorter computational time. Here we demonstrate the use of a spin-3/2 ( 7 Li) and spin-7/2
(133 Cs) nuclei in molecules oriented in liquid crystal matrices as 2- and 3-qubit systems,
respectively. We create pseudopure states, implement logic gates and half-adder and
subtractor logics using transition-selective pulses. While the feasibility of using these systems as qubits have been demonstrated earlier, very little computations were performed
[47,49].
3.1 2-qubit system using 7 Li (I = 3=2)
Figure 3b shows the three lines of 7 Li NMR spectrum (1.15 kHz splitting) of LiBF 4 oriented in a liquid crystal. This spectrum obtained at 310 K and at 194.37 MHz in a Bruker
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DRX-500 NMR spectrometer results from the quadrupolar-perturbed four Zeeman energy
levels of a spin-3/2 system, shown schematically in figure 3a. The equilibrium spectrum
has an intensity ratio of 1:1.6:1 corresponding to equal population differences between the
adjacent energy levels (under high temperature and high field approximations) which are
schematically shown on the rhs of figure 3c.
Pseudopure states in this system were earlier created using double quantum pulses [47].
We have instead created pseudopure states using transition-selective single quantum pulses,
which are much easier to create and implement [48]. Figures 3c–g summarize the creation
of various pseudopure states. The equilibrium spectrum and the schematic representation
of equilibrium population distribution are shown in figure 3c. The j11i pseudopure state
is created by applying a π (1 $ 2), which interchanges the populations of levels 1 and 2,
followed by a π =2(0 $ 1), which equalizes the populations of levels 0 and 1 to an average
value, followed by a gradient pulse which destroys the coherences created by π =2(0 $ 1).
This results in the pseudopure state j11i (figure 3d). Similarly other pseudo- pure states
have been created using transition-selective π and (π =2+ gradient) pulses as shown in
figure 3.
Figure 3. (a) Schematic energy level diagram for a spin-3/2 system and (b) equilibrium
7 Li NMR spectrum of LiBF oriented in a liquid crystalline matrix at 310 K, obtained
4
using a hard π =2 pulse with a Bruker DRX-500 NMR spectrometer. Equilibrium spectrum obtained using a small flip angle pulse (c) and spectra corresponding to different
pseudo-pure states (d–g). All spectra are shown in the same scale. Integrated intensities
of each peak are shown below each spectrum. The population distribution in each case
is shown on the rhs. The pulse sequence used for creation of the pseudopure state in
each case is shown on the lhs.
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Use of quadrupolar and dipolar couplings
Figure 4. 7 Li spectra corresponding to 24 one-to-one logic gates. These gates are
implemented using a non-selective inversion pulse (π ) on all the transitions and a ωi j
transition-selective inversion pulse. The unitary operators and the pulse sequences corresponding to these gates are given in ref. [48].
Figure 4 gives a complete set of the 24 reversible one-to-one two-qubit gates implemented using transition-selective pulses. The truth table, the circuit diagram and the pulse
sequence for each gate are given in ref. [48]. The spectra shown in figure 4 confirm the
implemented gate operations.
3.2 3-qubit system using
133 Cs (I = 7=2)
The equilibrium NMR spectrum of 133 Cs oriented in a lyotropic liquid crystal containing a mixture of cesium pentadecaflouro octonoate and D 2 O at 293 K recorded at 65.59
MHz in the DRX-500 NMR spectrometer is given in figure 5a. This seven-line equispaced
(quadrupolar splitting of 9.25 kHz) spectrum results from the equal population differences
of the eight energy levels. The equilibrium spectrum has theoretical intensities in the ratio of 7:12:15:16:15:12:7 due to differences in the matrix elements between the various
eigenstates of this (I = 7=2) spin system, matching with the integrated experimental intensities (within experimental errors). The eight eigenstates of this system can be treated
as a 3-qubit system [49–51] with a conventional labeling of the states in increasing energy as 7/2 j000i, 5/2 j001i, 3/2 j010i, 1/2 j011i; 1=2 j100i, 3=2 j101i,
5=2 j110i, and 7=2 j111i. We demonstrate implementation of the half-adder (HA)
and subtractor (SUB) logics using this system.
Half-adder and subtractor
Classically a half-adder (HA) operation is defined on two bits as the ‘sum’ = A B and
‘carry’ = A ^ B, where indicates addition modulo 2 and ^ indicates the ‘AND’ operation.
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Figure 5. (a) The equilibrium spectrum of 133 Cs of the lyotropic liquid crystal consisting of mixture of cesium pentadecaflouro octanoate and D2 O at 293 K recorded at
65.59 MHz in a 11.8 T (500 MHz) NMR spectrometer. The transitions are labeled in
increasing frequency, as indicated in table 3. Also shown schematically on the rhs are
the equilibrium populations of various energy levels. The spectra and the populations
after the implementations of (b) the half-adder (eq. (3)), (c) the subtractor (eq. (4)) and
(d) the C2 -NOT gate (implemented by a single transition-selective π pulse on transition 2).
Figure 6. Circuit diagrams of (a) a reversible half-adder and (b) a reversible subtractor.
In the case of the half-adder, B0 is the sum and C0 is the carry, and in the case of the
subtractor, C0 is the subtraction and A0 is the borrow. .
Table 1. Truth tables of an irreversible half-adder and an irreversible subtractor.
Input
A
B
0
0
1
1
0
1
0
1
Subtractor output
Half-output
Adder
Carry
A-B
Borrow
0
1
1
0
0
0
0
1
0
1
1
0
0
1
0
0
A+B
From the truth table of HA, given in table 1, it is noted that the input to output mapping
is many-to-one and is thus irreversible. Since quantum operations are carried out using
unitary operators, they have to be reversible. This reversibility can be achieved using an
ancillary bit. The truth table and the circuit diagram of the HA using an ancillary bit are
given in table 2 and figure 6a, respectively. In this scheme the bit C in the input and the bit
A in the output are redundant.
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Use of quadrupolar and dipolar couplings
Table 2. Input and output states of a reversible half-adder and a reversible
subtractor using the conventional labeling scheme.
ABC)
Output (
Transition
7f
6f
5f
4f
3f
2f
1f
Spin state
7/2
5/2
3/2
1/2
1/2
3/2
5/2
7/2
ABC)
Input (
111
110
101
100
011
010
001
000
0
0
0
Half-adder
Subtractor
100
101
111
110
011
010
001
000
110
111
001
100
010
011
101
000
The classical subtractor operation using two bits is also irreversible (table 1) and can be
made reversible using an ancillary bit (table 2 and figure 6b). Here the bit A in the input
and the bit B in the output are redundant.
Table 2 contains truth tables for a 3-qubit system having eight input-outputs, of which
only four are needed for both the half-adder and subtractor. The rest of four states with C=1
for HA and A=1 for subtractor and their outputs are not needed, but give complementary
carry and borrow outputs. In effect using complete 3-bit truth table of table 2, one obtains
the carry and borrow and their complement in half-adder and subtractor. The complement
may be useful in some other arithmetic operations. The HA operation given in table 2
can be carried out using a series of transition-selective inversion (π ) pulses given by the
sequence
HA = π7 π6 π5 π7 π6 ;
(1)
where the subscripts refer to the transition numbers given in table 2 (column 1) and figure
5a. The order of pulses read from right to left in eq. (1), such that first pulse is π 6 . The
output of the HA is given in column 4 of table 2.
The subtractor needs 9 transition-selective pulses in the sequence
subtractor
= π 3 π2 π3 π5 π4 π3 π2 π5 π7 ;
(2)
yielding the output given in column 5 of table 2.
The pulse sequences (for HA and subtractor) given in eqs (1) and (2) are both optimal (minimum pulses) for the present labeling. The number of transition-selective pulses
needed for the implementation of the HA and subtractor logics can be decreased if we
adopt a labeling scheme of the energy levels different from the one given in table 2. The
labeling scheme of table 2 labels energy levels in increasing energy. However, there is
no compelling argument in favor of this labeling scheme. Any labeling scheme should be
acceptable so long as each energy level is attached with one distinct label, and this label
is not changed during the entire set of computations. We have found that by labeling the
states which interchange under a set of operations (i.e., are connected by a single quantum
transition) sequentially, we can reduce the number of pulses needed for carrying out a set
of operations.
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Table 3. Input and output states of a half-adder and a subtractor using an optimized
labeling scheme. The circuit diagrams of the half-adder and subtractor are shown in
figure 6. The output of the Toffoli (C2 -NOT) gate is A0 =A, B0 =B, C0 =C(A^B).
ABC)
Output (
Transition
Spin state
7/2
5/2
3/2
1/2
1/2
3/2
5/2
7/2
7f
6f
5f
4f
3f
2f
1f
ABC)
Input (
000
010
011
001
101
110
111
100
0
0
0
Half-adder
Subtractor
C2 -NOT
000
010
011
001
111
101
100
110
000
011
010
101
001
111
110
100
000
010
011
001
101
111
110
100
For example, 111 interchanges with 110 in HA and remains unchanged in subtractor.
101 interchanges with 001 in HA and 011 with 010 in subtractor. Thus by placing 100,
111, 110, 101, 001, 011, 010 and 000 in an increasing order as shown in table 3, one can
execute the HA logic by the following operator,
HA = π1 π3 π2 ;
(3)
and the subtractor by
subtractor = π2 π4 π6 :
(4)
This leads to a minimum number of single quantum transition-selective π pulses for implementation of these logic operations. While we believe that the labeling scheme of table
3 is optimum (needs the minimum number of single quantum transition-selective π pulses)
for implementation of the HA and subtractor operations, it is by no means unique. This
labeling scheme has been arrived at by trial and error. However, for larger number of qubits
and for a larger set of computations the search for the optimal labeling scheme will need
to follow some systematic logic [51].
The experimental implementation of the HA and subtractor operations using the labeling
scheme of table 3 and pulse sequences of eqs (3) and (4) are given in figures 5b and 5c,
respectively. The final population differences are shown schematically on the rhs and the
corresponding spectra on the lhs of figure 5. The modified populations are the results of
three unitary transforms corresponding to eqs (3) and (4). The observed spectra of figure 5b
qualitatively confirms the population distribution shown schematically on the rhs of figure
5b. Similarly, it can be shown that the spectra of figure 5c corresponds to the population
distribution given in the rhs of figure 5c confirming the operation of the subtractor. It may
be mentioned that π 3 and π1 in the half-adder and all the pulses in the subtractor commute
and can be applied simultaneously using a modulator. The HA then reduces to two noncommuting operations, and the subtractor to a single operation. Table 3 also contains the
truth table for a C 2 -NOT operation, and the corresponding spectrum obtained using a single
transition-selective π pulse is shown in figure 5d.
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4. Conclusions
This paper demonstrates that dipolar and quadrupolar coupled nuclei can be used for logical operations and eventually for quantum information processing (QIP). The hope and
aim of the present work is that using such systems one can increase the number of qubits
in NMR based QIP. All operations have been carried out using single quantum transitionselective radio frequency pulses.
Acknowledgements
We thank Dr. J P Bayle, Universite de Paris-Sud, Orsay, France for providing the liquid
crystals used in this work. We also thank Prof. Malcolm Levitt for useful discussions. The
use of DRX-500 FTNMR spectrometer of the Sophisticated Instruments Facility, Indian
Institute of Science, Bangalore funded by the Department of Science and Technology, New
Delhi, is also gratefully acknowledged.
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